## Question 2

If P and Q are positive integers, is the product 3P^{Q} divisible by 2?

- 6Q
^{3}+ 2 is an even number - P + 8Q
^{2}is a prime number

### Correct Answer

B

### Solution

**Steps 1 & 2: Understand Question and Draw Inferences**

3P^{Q }is divisible by 2, if:

- 3P
^{Q }is even

–> P^{Q} is even (Odd term 3 plays no role in the even-odd nature of product 3P^{Q})

–> P is even (Power doesn’t impact the even-odd nature of a term)

So, to answer the question we need to find if P is even

**Step 3: Analyze Statement 1**

6Q^{3} + 2 is an even number

Not Sufficient. We do not know if P is even or odd

**Step 4: Analyze Statement 2**

P + 8Q^{2} is a prime number

All the prime numbers except 2 are odd

–> As Q ≠0, P + 8Q^{2}> 2 (Given: Q is a positive integer => Q >0)

–> P + 8Q^{2} is always odd

8Q^{2} is always even

–> P must be odd (Odd + Even = Odd)

Sufficient.

**Step 5: Analyze Both Statements Together (if needed)**

We get a unique answer in step 4, so this step is not required

**Answer: Option (B)**